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“Exploring” spherical-wave reflection coefficients
Chuck Ursenbach Arnim Haase Research Report: Ursenbach & Haase, “An efficient method for calculating spherical-wave reflection coefficients” Chuck Ursenbach
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Outline Motivation: Why spherical waves?
Theory: How to calculate efficiently Application: Testing exponential wavelet Analysis: What does the calculation “look” like? Deliverable: The Explorer Future Work: Possible directions
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Motivation Spherical wave effects have been shown to be significant near critical angles, even at considerable depth See poster: Haase & Ursenbach, “Spherical wave AVO-modelling in elastic isotropic media” Spherical-wave AVO is thus important for long-offset AVO, and for extraction of density information
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Outline Motivation: Why spherical waves?
Theory: How to calculate efficiently Application: Testing exponential wavelet Analysis: What does the calculation “look” like? Deliverable: The Explorer Future Work: Possible directions
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Spherical Wave Theory One obtains the potential from integral over all p: Computing the gradient yields displacements Integrate over all frequencies to obtain trace Extract AVO information: RPPspherical Hilbert transform → envelope; Max. amplitude; Normalize
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Alternative calculation route
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Outline Motivation: Why spherical waves?
Theory: How to calculate efficiently Application: Testing exponential wavelet Analysis: What does the calculation “look” like? Deliverable: The Explorer Future Work: Possible directions
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Class I AVO application
Values given by Haase (CSEG, 2004; SEG, 2004) Possesses critical point at ~ 43 Upper Layer Lower Layer VP (m/s) 2000 VS (m/s) 879.88 r (kg/m3) 2400
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Test of method for f (n) = n4 exp([.173 Hz-1]n)
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Wavelet comparison
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Spherical RPP for differing wavelets
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Outline Motivation: Why spherical waves?
Theory: How to calculate efficiently Application: Testing exponential wavelet Analysis: What does the calculation “look” like? Deliverable: The Explorer Future Work: Possible directions
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Behavior of Wn qi = 15 qi = 0 qi = 45 qi = 85
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Outline Motivation: Why spherical waves?
Theory: How to calculate efficiently Application: Testing exponential wavelet Analysis: What does the calculation “look” like? Deliverable: The Explorer Future Work: Possible directions
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Pseudo-linear methods
Spherical effects Waveform inversion Joint inversion EO gathers Pseudo-linear methods
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Conclusions RPPsph can be calculated semi-analytically with appropriate choice of wavelet Spherical effects are qualitatively similar for wavelets with similar lower bounds New method emphasizes that RPPsph is a weighted integral of nearby RPPpw Calculations are efficient enough for incorporation into interactive explorer May help to extract density information from AVO
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Possible Future Work Include n > 4
Use multi-term wavelet: Sn An wn exp(-|snw|) Layered overburden (effective depth, non-sphericity) Include cylindrical wave reflection coefficients Extend to PS reflections
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Acknowledgments The authors wish to thank the sponsors of CREWES for financial support of this research, and Dr. E. Krebes for careful review of the manuscript.
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